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We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic… (More)

We present the first complete, exact and efficient C++ implementation of a method for parameterizing the intersection of two implicit quadrics with integer coefficients of arbitrary size. It is based on the near-optimal algorithm recently introduced by Dupont et al., [2]. Unlike existing implementations, it correctly identifies and parameterizes all the… (More)

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic… (More)

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an… (More)

We summarize the results of the cross-benchmarks for two univari-ate algebraic kernels (AK) developed in ACS. The kernels, developed at INRIA and MPI, were tested on 6 types of univariate polynomials of various degrees and bitsizes. The methods included were: Sturm, Sleeve, CF, NCF, NCF2 for the INRIA kernel, and Descartes and Bitstream-Descartes for the… (More)

We present a cgal-based univariate algebraic kernel, which provides certied real-root isolation of univariate polynomials with integer coecients and standard functionalities such as basic arithmetic operations, greatest common divisor (gcd) and square-free factorization, as well as comparison and sign evaluations of real algebraic numbers. We compare our… (More)

We design an algorithm to compute the Newton polytope of the resultant , known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex-and halfspace-representations of the polytope using an oracle producing… (More)

Solving univariate polynomials and multivariate polynomial systems is critical in geometric computing with curved objects. Moreover, the real roots need to be computed in a certified way in order to avoid possible inconsistency in geometric algorithms. We present a Cgal-based univariate algebraic kernel, which follows the Cgal specifications for univariate… (More)

Determinant computation is the core procedure in many important geometric algorithms, such as convex hull computations and point locations. As the dimension of the computation space grows, a higher percentage of the computation time is consumed by these predicates. In this paper we study the sequences of determinants that appear in geometric algorithms. We… (More)

The Orientation predicate is the bottleneck of some important geometric algorithms, such as convex hull and triangulation computations, since it is computed repeatedly. In these and other applications, the matrices whose determinants are computed along the execution of an algorithm have many columns in common. Besides Orientation, this concerns all… (More)